rem-1.2

Let $M$ be a Hausdorff topological space and $E:\mathbb R\to\operatorname{Opens}(M)$ a family of open sets satisfying: $E(t)\subseteq E(s)$ for $t\le s$; $E(t)=\operatorname{int}_M(\bigcap_{s>t}E(s))$ for every $t$; $\operatorname{int}_M(\bigcap_t E(t))=\varnothing$; $\bigcup_t E(t)=M$; and $\overline{E(t)}\subseteq E(s)$ whenever $t<s$. Let $x\in M$ satisfy $\exists t,\ x\notin E(t)$. Let $B$ be a nonempty family of open sets excluding $\varnothing$, with a common smaller member for every pair of members, maximal under inclusion among such families, and suppose that $\bigcap_{U\in B}\overline U=\{x\}$. Then the set $A_B=\{t\in\mathbb R:E(t)\in B\}$ is nonempty and bounded below, and

Both infima are taken in $\mathbb R$, whose total infimum operation returns $0$ on an empty set or a set unbounded below. The spectral conditions involving arbitrary meets are the interior-of-intersection conditions displayed above, rather than assertions that the corresponding intersections themselves are open or empty.